1. Field of the Invention
This invention relates generally to computer software programs for modeling water distribution systems, and more particularly, to incorporating pressure dependencies in such models.
2. Background Information
A computer model of a water distribution system (also referred to herein as a hydraulic network) is a helpful tool to engineers involved in design, maintenance and day to day operation of the water distribution system. The model is created by representing the system as a network of links and nodes. The links are the pipes in the system. A node is a point in the network where water consumption is calculated and allocated as “demand.” A node may be a household or a building in the network, or an arbitrary point of interest selected by the modeling engineer. Storage tanks, valves and pumps are also taken into account in the computer model of the network. The network behavior is emulated by the computer model using a link-node formulation that is governed by two conservation laws, namely mass balance at nodes, i.e., the mass of water coming into the node is equal to the mass of water exiting the node, and energy conservation around hydraulic loops, i.e., the total energy in the loop is constant. In conventional models, demand is treated as a known value. In other words, when initializing a hydraulic analysis model, typically, data is provided as to the customer's demand values. As long as the pressures are relatively constant in the system as they usually are in a normally running system in most developed nations, then that is adequate. However, the pressure may widely vary during the day as in a developing nation, or when there are special conditions in parts of the system such as pipe breaks, or power outages or fire fighting. In such cases, the assumption is that the demands are constant is not true. Instead, demand varies with pressure. Accounting for these conditions requires calculating pressure dependent demand.
Conventional water distribution models are formulated under the assumption that water consumptions (referred to as “demands”) defined at nodes are known values so that nodal hydraulic heads and pipe flows can be determined by solving a set of quasi-linear equations. As noted, the network behavior is emulated by the computer model using a link-node formulation governed by the above mentioned conservation laws, namely mass balance at nodes, and energy conservation around hydraulic loops. More specifically, a traditional model is “solved” when a set of network hydraulics solutions are obtained by iteratively solving a set of linear and quasi-nonlinear equations that are governed by mass conservation law to each node and energy conservation law to each loop. Thus, as there is an equation for each node and each loop, these equations are solved using matrix techniques. Using the known demand value, the nodal hydraulic head is determined. As is well known to those skilled in the art, hydraulic head is a measurement of total energy per unit weight at a particular height. The quantity of water flowing from one node to another node should be equal, but the rate of flow from the node of upper height to the node of lower height is defined by a hydraulic gradient. The hydraulic gradient also provides a measurement related to the demand. One problem being solved by the present invention relates to formulating the correct relationship (equation) for demands and pressure dependencies of such demands, and to solving such equations to efficiently calculate accurate pressure dependent demand values and flows at given nodes in the hydraulic network.
In one prior known approach, a unified formulation for solving network hydraulics was provided as a “global gradient algorithm” (GGA) (Todini & Pilati 1988 and Todini 1999) which is set forth in the following matrix equation:
                                          [                                                                                A                    11                                                                    …                                                                      A                    12                                                                                                …                                                  …                                                  …                                                                                                  A                    21                                                                    …                                                  0                                                      ]                    ⁡                      [                                                            Q                                                                              …                                                                              H                                                      ]                          =                  [                                                                                          -                                          A                      10                                                        ⁢                                      H                    0                                                                                                      …                                                                                      -                  q                                                              ]                                    (        2        )            
Where Q is the [np, 1] unknown pipe discharges in such units as feet per second; H represents the [nn, 1] unknown nodal heads measured in units of length such as feet, and q represents the [nn, 1] known nodal demands measured in units of volume such as cubic feet; H0 is the [nt−nn, 1] known nodal heads; A11 is a [np, np] diagonal matrix for pipes and pumps; A12 and A21 are the [np, nt] topological incidence matrix that defines the pipe and node connectivity; A10 is a [nt−nn, 1] topological incidences for known-head nodes; nt is the total number of nodes and nn is the number of unknown-head nodes and np is the total number of links.
This formulation can be used to solve for the demand (q), but it is valid only if the hydraulic pressures at all nodes are adequate so that the demand is independent of pressure. It is also valid approach for analyzing volume-based demand such as filling bath tubs, flushing toilets etc. even under low pressure conditions. However, there are a number of scenarios where nodal pressure is not sufficient for supplying the required demand. As noted, these cases may include planned system maintenance, unplanned pipe outages, power failure at pump stations, fire fighting and insufficient water supply from water sources. In addition, some water consumptions like leakages are pressure dependent. In is such scenarios, where there is a pressure dependency, the conventional hydraulic modeling techniques are insufficient and possibly inaccurate for correctly modeling the behavior of the network under these pressure dependent circumstances.
Correctly measuring and predicting pressure dependent demand is an important requirement for companies providing water to communities because such companies must constantly evaluate the level of water supply service while coping with emergency events. In the United Kingdom (UK), for example, a tentative guideline requirement (Ofwat 2004) is that a water system must meet a level of the original demand for the majority of customers such that the water companies are required by law to provide water at a pressure that will, under normal circumstances, enable it to reach the top floor of a house. In order to assess if this requirement is satisfied, the water companies are required to report against a service level corresponding to a pressure head of 10 meters (14.2 psi) at a flow of 9 liter per minute (2.4 gpm). In addition, the supply reference for unplanned and planned service interruptions is also to be evaluated and reported.
Thus, water asset management has become an ever-increasing task for water utilities. It requires a comprehensive evaluation of above and underground facilities including every pipeline segment in a water system. Thus, impact and criticality analyses are carefully performed for each pipe segment in a system in order to form a rational basis for asset management plan. The impact evaluation is usually undertaken by performing the hydraulic analysis under the assumption that a pipe or a number of pipes is out of service, namely disconnected from a system, which is likely to cause pressure deficient conditions. The accurate analysis cannot be achieved without considering the impact of the pressure change on the flow supplied. is There remains a need, therefore, for a water distribution modeling system that can be used to efficiently calculate pressure dependent demand and to thus accurately predict network behavior under various scenarios.